PHYSICAL REVIEW E 76, 046608 2007

Two-dimensional discrete solitons in rotating lattices
Grupo de Física No Lineal, Departamento de Física Aplicada I, Escuela Universitaria Politécnica, C/ Virgen de África, 7, 41011 Sevilla, Spain 2 Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel 3 Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA Received 15 May 2007; published 19 October 2007 We introduce a two-dimensional discrete nonlinear Schrödinger DNLS equation with self-attractive cubic nonlinearity in a rotating reference frame. The model applies to a Bose-Einstein condensate stirred by a rotating strong optical lattice, or light propagation in a twisted bundle of nonlinear ﬁbers. Two types of localized states are constructed: off-axis fundamental solitons FSs , placed at distance R from the rotation pivot, and on-axis R = 0 vortex solitons VSs , with vorticities S = 1 and 2. At a ﬁxed value of rotation frequency , a stability interval for the FSs is found in terms of the lattice coupling constant C, 0 C ˜ S=1 C Ccr R , with monotonically decreasing Ccr R . VSs with S = 1 have a stability interval, Ccr S=1 S=1 Ccr , which exists for below a certain critical value, cr . This implies that the VSs with S = 1 are destabilized in the weak-coupling limit by the rotation. On the contrary, VSs with S = 2, that are known to be S=2 unstable in the standard DNLS equation, with = 0, are stabilized by the rotation in region 0 C Ccr , with S=2 Ccr growing as a function of . Quadrupole and octupole on-axis solitons are considered too, their stability regions being weakly affected by 0. DOI: 10.1103/PhysRevE.76.046608 PACS number s : 05.45.Yv, 03.75.Lm, 42.65.Tg, 42.70.Qs
1

Jesús Cuevas,1 Boris A. Malomed,2 and P. G. Kevrekidis3

I. INTRODUCTION

Discrete dynamical systems represented by nonlinear lattices in one, two, and three dimensions constitute a class of models which are of fundamental interest by themselves, and, simultaneously, they ﬁnd applications of paramount importance in various ﬁelds of physics. One such example is known in nonlinear optics, where the one-dimensional 1D discrete nonlinear Schrödinger DNLS equation was predicted by Christodoulides and Joseph to support fundamental discrete solitons 1 with an even proﬁle. Such nonlinear structures were later created experimentally in an array of semiconductor waveguides 2 . Subsequently, stable odd solitons, alias twisted localized modes, were predicted and studied in detail in the same model 3 , as well as in an array of photorefractive waveguides with photovoltaic nonlinearity 4. Another experimental realization of dynamical lattices in the optical domain is possible in photorefractive crystals, where a quasidiscrete setting can be induced by counterpropagating laser beams illuminating the crystal in the normal polarization, while the probe beam, which can sustain solitary waves, is launched in the extraordinary polarization. The difference from the array of waveguides fabricated in silica or in a semiconductor material is that the photorefractive nonlinearity is saturable, rather than cubic. This method of the creation of photonic lattices was proposed in Ref. 5 , and results obtained by means of the technique were reviewed in Ref. 6 . In particular, both fundamental and twisted solitons in the 1D lattice were reported in Ref. 7 , and fundamental solitons FSs in the 2D lattice were created too 8 , as well as 2D vortex solitons VSs in the same setting 9 . Recently, the progress in the technology of writing permanent arrays of channels in silica slabs has made it possible to
1539-3755/2007/76 4 /046608 12

lar, 2D solitons of both the fundamental and vortex types, which are unstable in uniform continua with the cubic selffocusing nonlinearity, can be readily stabilized by the periodic OL potential 22 ; for the stabilization of FSs, a quasi-1D potential is sufﬁcient, instead of its full 2D counterpart 23 . Vortices are unstable too in the uniform space with the saturable nonlinearity 24 , in which case they can also be stabilized by the periodic potential 22,9 . The relevance of these stabilization mechanisms in the continuum was also demonstrated for higher-order vortex solitons, and so-called supervortices, i.e., arrays of compact vortices with global vorticity imprinted onto the array, under both cubic and saturable nonlinearities 25 . Recently, it was shown that 2D solitons obeying the GPE in the 2D continuum can also be supported by a rotating OL 26,27 . These solitons may be fully localized spot-shaped solutions to the equation with the self-focusing or attractive cubic nonlinearity, placed at some distance from the rotation pivot and revolving in sync with the holding 2D lattice. In particular, the soliton can be placed at a local minimum of the rotating potential, while the pivot is set at a local maximum. These co-rotating strongly localized solitons are stable provided that the rotation frequency does not exceed a critical value cr min. In the same model, but with a rapidly rotating OL, stable ring-shaped solitons with zero vorticity , i.e., objects localized along the radius but delocalized in the azimuthal direction, have been found too, for exceeding another critical value cr max. Note that the model does not support any stable pattern in interval cr min cr max 26 . On the other hand, stable ring-shaped states, with both zero and nonzero vorticity, have been found in the model with the repulsive cubic nonlinearity and rotating quasi-one-dimensional periodic potential, if exceeds a respective critical value. Obviously, the latter model does not give rise to any localized state in the absence of the rotation. A rotating OL can be easily implemented in BEC experiments 28 ; then, if the OL is strong enough, it is natural to approximate the GPE in the co-rotating reference frame by an appropriate variety of the 2D DNLS equation. In addition to that, such a model may also describe the light propagation in a twisted bundle of nonlinear optical ﬁbers, linearly coupled in the transverse plane by tunneling of light between the ﬁber cores. The objective of the present work is to introduce a model of the rotating discrete lattice, and ﬁnd stable discrete solitons in it, both FSs and VSs. The paper is organized as follows. In Sec. II, we formulate the model, taking the underlying GPE in the reference frame co-rotating with the OL, and replacing the continuum equation by its discrete version corresponding to a strong periodic potential. Discrete FSs are considered in Sec. III. We construct the solutions starting from the anticontinuum limit, which corresponds to the zero value of the coupling constant accounting for the linear interaction between neighboring sites of the discrete lattice, C = 0. A family of FS solutions is constructed by continuation in C; their stability is examined by the computation of eigenfrequencies for inﬁnitesimal perturbations around the soliton, and veriﬁed by direct simulations of the evolution of perturbed FSs. It is found that the FS, with its center located at distance R from the rotation pivot, is stable within an interval 0 C Ccr R ,

with Ccr decaying as a function of R. Section III also includes a simple analytical approximation, which makes it possible to explain the decrease of Ccr with the growth of R. In Sec. IV, we consider localized vortices VSs , whose center coincides with the rotation pivot. For the VS with vortic˜ S=1 C ity S = 1, the stability region is found to be Ccr S=1 Ccr , provided that the rotation frequency is smaller than a critical value cr the stability interval shrinks to nil at = cr . Vortices with S = 2 are considered too. While in the ordinary nonrotating DNLS model, with = 0, all VSs of the latter type are unstable 29 , we demonstrate that the S=2 rotation opens a stability window for them, 0 C Ccr , S=2 with Ccr growing as a function of . Direct numerical simulations are also used to illustrate the dynamical evolution of FSs and VSs when they are unstable. Results obtained in this work and related open problems are summarized in Sec. V.
II. MODEL

The starting point is the normalized 2D GPE, which includes the potential in the form of an OL rotating at angular velocity , and thus stirring a “pancake”-shaped quasi-ﬂat Bose-Einstein condensate BEC trapped in a narrow gap between two strongly repelling optical sheets. Unlike the analysis performed in Refs. 26,27 , where simulations were run in the laboratory reference frame, here we write the GPE in the reference frame co-rotating with the lattice, hence the potential does not contain explicit time dependence: 1 2
2

i

t

=− +

+ .

ˆ Lz

−

cos„k x −

… + cos„k y −

… 1

2

ˆ Here, Lz = i x y − y x i is the operator of the z component of the orbital momentum is the polar angle , and determines the sign of the interaction, attractive = −1 or repulsive = + 1 . Constants and determine a possible shift of the lattice with respect to the rotation pivot. Note that Eq. 1 does not contain any additional trapping potential, as we are interested in solutions localized under the action of the OL. It is more convenient to shift the origin of the Cartesian coordinates to a lattice node, thus replacing Eq. 1 by the translated form 1 2
2

i

t

=− −

+i

x+

y

− y+ +
2

x

cos kx + cos ky

.

2

As shown in a general form in Ref. 14 , a discrete model, which corresponds to the limit of a very deep OL, can be derived from the underlying GPE in the tight-binding approximation. Eventually, it amounts to a straightforward discretization of the GPE. Thus, the discrete counterpart of Eq. 2 is

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i

d

m,n

dt

=−

C 2

m+1,n

+

m−1,n

+

m,n+1

+

m,n−1

−4

m,n

To parametrize the soliton families, we ﬁx the scales by setting 1; obviously, with C 0, only positive may give rise to localized solutions, while C will be varied. 3
III. FUNDAMENTAL SOLITONS

−i − n+

m+

m,n+1 m+1,n

−

m,n−1

−

m−1,n

+

m,n

2

m,n ,

where m , n are discrete coordinates, and C 0 is the corresponding coupling constant, which accounts for the linear tunneling of atoms between BEC droplets trapped in deep nodes of the lattice. As mentioned above, Eq. 3 may also describe a twisted bundle of nonlinear optical ﬁbers linearly coupled by the tunneling of light in the transverse plane, m , n . In that case, t is the propagation distance along the ﬁber, and only = −1, i.e., the attractive or focusing nonlinearity, is the relevant choice. Equation 3 conserves the norm and Hamiltonian, N=
m,n m,n 2

,

4

H=
m,n

C 2

m+1,n

− „

m,n

2

+

m,n+1

−

m,n

2

+

1 2

m,n

4

The rotation makes the discrete lattice inhomogeneous, hence the properties of solitons strongly depend on the location of their centers. Without the rotation, Eq. 3 amounts to the ordinary 2D DNLS equation, in which various species of discrete solitons and their stability have been studied in detail. In particular, FSs, which are represented by real solu2 21 . The onset of their tions, are stable at C Ccr = 2 instability is accounted for by a pair of eigenfrequencies of small perturbations with ﬁnite imaginary and zero real parts, i.e., the instability leads to the exponential growth of perturbations. Accordingly, numerical simulations of the instability development demonstrate spontaneous transformation of unstable FSs into lattice breathers 30 . In this section, we ﬁrst report numerical results obtained for the stability of FSs in the model with 0, and then present an analytical estimate that may explain numerical ﬁndings.
A. Numerical results

− − −

iC 4
m,n„ m,n

m+
* m,n+1 * m+1,n

* m,n

m,n+1

−

m,n−1

− −

* m,n−1…… * m−1,n

− n+ .

„

* m,n

m+1,n −

m−1,n

…

5

In addition to C, the discrete model contains three irreducible parameters: , which takes values 0 , and the coordinates of the pivot displacement, , , which take values , 1, plus the sign parameter, = ± 1. As in the usual 0 2D DNLS equation with = 0 , the values = ± 1 in Eq. 3 may be transformed into each other by the staggering trans−1 selfformation m,n → −1 m+n m,n, therefore we ﬁx attraction . Our ﬁrst objective is to ﬁnd stationary localized solutions to Eq. 3 in the form of FSs and VSs. To this end, we substitute the standing wave ansatz m,n = ei t m,n, where − is the normalized chemical potential, in terms of the underlying BEC model; then, the stationary lattice ﬁeld m,n obeys the equation
m,n

=

C 2 +i C 2

m+1,n

+

m−1,n

+

m,n+1

+

m,n−1

−4

m,n

m+
m+1,n

m,n+1

−

m,n−1

− n+

−

m−1,n

+

m,n

2

m,n .

6

Note that solutions for m,n are complex, unless = 0. The second objective will be to examine the stability of the discrete solitons, assuming small perturbations in the form of m,n exp i t + i t , the onset of instability indicated by the emergence of Im 0. The evolution of unstable solitons will be examined by means of direct simulations of Eq. 3.

Our analysis aimed to determine the stability border for the FSs, Ccr, for each set of values of the discrete coordinates of the soliton’s center, m0 , n0 . Here we present results for angular velocity = 0.1 and zero pivot displacement = = 0 in Ref. 26 , the rotation pivot was ﬁxed at a local maximum of the potential in Eq. 2 , which corresponds to setting = = 1 / 2 in Eqs. 3 and 6 , and the center of the soliton trapped in the lattice was placed at a local minimum closest to the pivot, which would mean m0 , n0 = 0 , 0 . This choice makes it possible to explore the existence and stability of FSs in a clear form, while larger values of give rise to a resonance with linear lattice modes, leading to Wannier-Stark ladders and hybrid solitons 33 and making the continuation in C and identiﬁcation of Ccr difﬁcult. We carried out the calculations on the lattice of size 21 21, since for this case, the lattice was for all the considered cases much wider than the very localized FS structures of interest. To avoid effects of the boundaries, the range of the soliton center’s coordinates was restricted to m0 , n0 8. The FS solutions were looked for starting at point C = 0, i.e., at the so-called anticontinuum limit 21 . In this limit, the FS is seeded by using a nonzero value of the ﬁeld at a C=0 single point, the center of the FS, m,n = m,m0 n,n0; obviously, this expression satisﬁes Eq. 6 with = 1 and C = 0. After the branch of the FS solutions had been found by the continuation in C, its stability was quantiﬁed through the computation of eigenfrequencies of small perturbations, using the equation linearized about the FS. Figure 1 displays a typical example of the thus found dependence of eigenfrequencies of small perturbations on the lattice coupling constant as said above, the instability corresponds to Im 0 , for the FS with its center set at point m0 = 3 , n0 = 2 ; for comparison, the dependence of the instability growth rate, i.e., Im , on C is also shown for

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PHYSICAL REVIEW E 76, 046608 2007

|Re(λ)|

|Im(λ)|

6 5 4 3 2 1 0 0.5 1 1.5 2

0.4 0.3 0.2 0.1 0 0

C

0.5

1

C

1.5

2

9 8 7 6
0.7 0.6 0.5

|Re(λ)|

5 4 3 2 1 0 0.5 1 1.5 2

|Im(λ)|

0.4 0.3 0.2 0.1 0 0

C

0.5

C

1

1.5

2

FIG. 1. Color online Plots of real and imaginary parts left and right panels of the eigenfrequencies of linearization around fundamental solitons in the ordinary nonrotating lattice, with = 0 top panels , and for their counterparts in the present model with = 0.1 . In the latter case, the soliton is centered at m0 = 3 , n0 = 2 .

the FS in the usual DNLS model = 0 in the top panel of the ﬁgure. It is seen that the instability sets in at C = Ccr = 1.70, which is smaller than the critical value in the ordinary model Ccr =0 = 2. Typical examples of stable and unstable FSs belonging to the family presented in Fig. 1 are displayed in Figs. 2 and 3. Results obtained for the FSs placed at different positions are summarized in Fig. 4, in the form of dependences of Ccr on the distance of the FS’s center from the pivot, R m2 + n2, and on one coordinate n0, while m0 is ﬁxed. It is 0 0 observed that Ccr monotonously decreases with R, starting from Ccr = 2 at R = 0 we recall again that Ccr =0 = 2 for the FS in the ordinary DNLS equation . Note that there are different pairs m0 , n0 which have equal values of R = m2 + n2 and 0 0 give slightly different Ccr. For instance, for the pair 5 , 0 , Ccr = 1.51, while Ccr = 1.50 for 4 , 3 . Hence, the stability depends on the two-dimensional structure of the solution. Figures 1 and 3 show that the FS is destabilized, with the increase of C, through the appearance of a pair of imaginary eigenfrequencies. Direct simulations of the dynamical evolu-

tion of unstable FSs in the framework of Eq. 3 demonstrate that the instability does not destroy the solitary wave. Instead, it transforms the waveform into a persistent breathing structure see a typical example in Fig. 5 .
B. Semi-analytical estimates

The decrease of Ccr with the increase of the distance of the FS from the pivot R, which is the main feature revealed by the above numerical analysis, as shown in Fig. 4, can be explained using an estimate based on the quasicontinuum approximation. To this end, we note that stationary solutions to the underlying continuum equation 1 are looked for as = ei t x , y , with the function obeying the stationary equation =−C 1 2
2

+

ˆ Lz

−

cos kx + cos ky

−

3

. 7

Here, following the discrete model, we have set = = 0, = −1, and explicitly introduced the spatial-scale parameter

FIG. 3. Color online The same as in Fig. 2, but for an unstable fundamental soliton, placed at the same position as in Fig. 2, at C = 1.8.

1 / C, which is a counterpart of the lattice coupling constant in Eq. 3 . Next, we assume the presence of a soliton with amplitude A and intrinsic size l, whose center is located at distance R from the rotation pivot tantamount to the origin, in the present case . First, demanding a balance between

, 2 , and 3 in Eq. 7 , and estimating them, terms respectively, as A, A / l2, and A3, we conclude, in the lowest approximation, that Cl−2 A2 . 8

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|ψ2,3|2

C

3.65 3.6 3.55 3.5 3.45 0

1.2 1 0.8

0

2

4

R

6

8

10

12

100

200

t

300

400

500

2 1.8 1.6 1.4

FIG. 5. Color online The time dependence of the squared amplitude of a perturbed unstable soliton from Fig. 3 at its center m0 = 3 , n0 = 2 . Notice the robust oscillatory behavior indicating the breathing nature of the resulting solution.

1.2 1 0.8

0

1

2

3

n

4 0

5

6

7

8

FIG. 4. Color online Dependence of the stability border for the fundamental solitons Ccr on the distance of the soliton’s center from the rotation pivot R top panel , and on one of the center’s coordinates n0 for ﬁxed m0 bottom panel . In the latter ﬁgure, the value of n0 0 , 8 increases progressively from top to bottom.

Using the quasicontinuum approximation, it is also possible to estimate the order of magnitude of the rotation frequency, in physical units. In the application to the BEC, assuming the lattice spacing 1 m and the condensate of 7Li or 85Rb, which provide for the possibility of the attraction between atoms and, thus, the formation of solitons 31,32 , and undoing of rescalings which cast the GPE in the normalized form of Eq. 1 , we conclude that = 0.1 corresponds, in physical units, to the rotation frequency 100 or 10 Hz, for lithium and rubidium, respectively. As concerns the above-mentioned realization of the model in terms of the twisted bundle of optical ﬁbers, an estimate shows that, for the carrier wavelength 1 m and separation between the ﬁbers in the bundle 10 m, = 0.1 corresponds to the twist pitch, which we deﬁne as a length at which the twist attains the angle of 2 , of the order of 5 cm.
IV. VORTEX SOLITONS

C

cr

Further, the soliton as a whole will be in equilibrium relative to the rotating lattice potential if the action of the centrifugal force, generated by the term in Eq. 7 , is compensated by the force of pinning to the periodic potential. . SubThe estimate of the latter condition yields C R / l C / , as per Eq. 8 , we arrive at a ﬁnal stituting here l / C−1/2, which predicts dependence estimate, R 2 Ccr 1 / R Ccr is realized here as the largest value of C that can provide for the balance between the centrifugal and pinning forces at given R . The latter dependence is qualitatively consistent with the numerical ﬁndings showing the decrease of Ccr with R in Fig. 4 at very large C, when the analytical formula predicts small R, it is irrelevant, as the above consideration tacitly assumed that the size of the soliton was essentially smaller than the distance to the pivot, l R; it is irrelevant too at very small C, as the quasicontinuum approximation cannot be used in that essentially discrete case .

Discrete VSs of the 2D DNLS equation were systematically developed in Ref. 35 as complex stationary solutions which feature a phase circulation of 2 S around the central point, at which the amplitude vanishes, with integer S identiﬁed as the vorticity. Prior to that, time-periodic multibreather states that may feature a vortical structure were found in 2D Hamiltonian lattice dynamical models 36 . The center of the VS may coincide with a site of the lattice, or may be located in the middle of a lattice cell; the corresponding vortices are called “crosses” alias rhombuses and “squares,” respectively. The stability of these states has been studied in detail, both for S = 1 35,37 and S 1 29,37 . In particular, the VS crosses with S = 1 and, as above, with ﬁxed to be 1 are stable in the corresponding interval of S=1 values of the coupling constant, C Ccr = 0.781, and the instability above this point transforms the VS into an ordinary fundamental soliton, with S = 0. While square VSs with S = 2 are unstable in the ordinary 2D DNLS equation, here

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the vortex solutions with S = 3 have their stability interval S=3 more narrow than for S = 1 , C Ccr = 0.198. In all cases, the instability sets in via the Hamiltonian Hopf bifurcation, represented by quartets of eigenvalues 38 . We have constructed localized vortices of the cross or rhombus type , with S = 1 and S = 2, in the rotating-lattice model based on Eq. 3 , and examined their stability. Unlike the above analysis of the FSs, we consider here only on-axis VSs, whose centers coincide with the rotation pivot R = 0 . We note that, while the FS with R = 0 is virtually identical to its counterpart in the ordinary model, with = 0, for VSs the situation is quite different; in particular, the VSs centered at the rotation pivot feature quite nontrivial stability properties with the increase of rotation frequency see below . Therefore, while the results for the FSs were displayed for = 0.1 and growing R, in this section we focus on R = 0 but vary .
A. Vortex solitons with S = 1

The ﬁrst noteworthy effect of the rotation on the VS with S = 1 is that, for given , its stability region features not only S=1 the upper bound Ccr , as in the usual DNLS model, but also S=1 ˜ a lower one, C
cr

˜ S=1 Ccr

C

S=1 Ccr .

9

At a given value of , VSs are exponentially unstable for ˜ S=1 C Ccr , and they feature an oscillatory instability, acS=1 counted for by a Hopf bifurcation, at C Ccr . This situaS=1 tion takes place for = 0.037. An example is given cr by Fig. 6, which displays the instability growth rate of the VS Im , versus C for ﬁxed = 0.01. The plot also shows the real part of the eigenfrequencies, Re , and includes, for the sake of comparison, the same dependences in the usual DNLS equation, with = 0. The ﬁgure shows that, in S=1 ˜ S=1 this case, Ccr = 0.31 and Ccr = 0.788. The overall stability region for the VSs with S = 1 in the S=1 plane C , is presented in Fig. 7. It is observed that Ccr ˜ S=1 slightly increases with , while the growth of Ccr with S=1 is fast. As a result, at the stability region does not cr exist. In the absence of the stable VSs, unstable ones feature coexistence of exponential and Hopf instabilities. Figure 6 displays an example of the latter regime, and shows that, for S=1 ˜ S=1 = 0.04, Ccr = 0.87 and Ccr = 0.81. In fact, the effect of the destabilization of the VSs with ˜ S=1 S = 1 at small C C Ccr , as mentioned above , is a higherorder phenomenon, in terms of the expansion of the stability eigenvalues in powers of C, when it is assumed small. Indeed, comparing it with the stability analysis for the cross vortices in the ordinary model, with = 0 39 , we observe the following. At = 0, modes of small perturbations around the vortex of the cross or rhombus type feature a pair of real eigenfrequencies, with = ± C at the ﬁrst order in small C in the present notation . The same pair appears in the present context see Fig. 6 . At larger C, these eigenfrequencies will give rise to the Hamiltonian-Hopf instability, upon their collision with eigenfrequencies bifurcating from the phonon

band of linear excitations. As shown in Ref. 39 , the DNLS equation with = 0 also gives rise to a pair of higher-order eigenfrequencies, ± C2 in the present notation. The effect of the rotation forces the latter eigenfrequencies to separate along the imaginary axis, thus inducing the instability at small C. However, for larger C, the pair again becomes real, stabilizing the conﬁguration. Four different regimes identiﬁed in the stability diagram in Fig. 7 are illustrated by typical examples of the stability and instability of the VSs with S = 1 in Fig. 8 as follows: the exponential instability, caused by the higher-order eigenfrequencies, as described above, is shown in the top left panel; the top right panel shows a linearly stable case. The oscillatory instability is presented in the bottom left panel the latter case is shown for sufﬁciently large C, to allow the eigenfrequencies, which originally linearly depend on C, as indicated above, to collide with eigenfrequencies bifurcating from the phonon band . Finally, the mixed oscillatory-exponential inS=1 , is displayed in the stability, which is possible at cr bottom right panel. S=1 Since the destabilization of the VS at C Ccr is accounted for by the Hopf bifurcation, as conﬁrmed by Fig. 6, this unstable VS is transformed into a persistent breather not shown here , which loses the vortical structure, i.e., is similar to the FS 35 . On the other hand, the destabilization at C ˜ S=1 occurs, as seen in Fig. 6, via a pair of imaginary Ccr eigenfrequencies with zero real parts, i.e., an exponential instability. Its nonlinear development eventually leads to a persistently pulsating localized state with zero vorticity, as illustrated by Fig. 9. The transformation to a FS state is also observed in the region of the coexistence of exponential and oscillatory instabilities.
B. Vortex solitons with S = 2

Another way in which the rotating lattice drastically alters the stability features of the ordinary 2D DNLS model concerns the VSs with S = 2. At = 0, all square localized vortices with S = 2 are unstable due to an imaginary eigenfrequency proportional at the leading order to coupling constant C 35,37 , see the top panel of Fig. 10 which may be compared with Fig. 4 of 37 . In the rotating lattice, the solitons with S = 2 acquire a ﬁnite stability region, as manifested by the example displayed in Fig. 10. Note that, unlike the stability interval for the VSs with S = 1, see Eq. 9 , only an upper stability border exists for the S = 2 solitons, i.e., the S=2 respective stability interval is 0 C Ccr . For instance, in the case shown in Fig. 10, the stability border induced by the S=2 rotation is Ccr = 0.12. Note that the mechanism of the stabilization of the S = 2 VSs in the present model is different from that reported in Ref. 40 , where localized vortices with S = 2 were stabilized by an impurity inert site placed at the center. In that case, the unstable eigenmode was suppressed by the defect, making all eigenfrequencies real; eventual destabilization occurred due to collisions of those real eigenfrequencies with the linear spectrum. We should also note that very recently rhombic vortices of S = 2 were proposed 41 and were found to be linearly stable for small C. Here, the rotation affects the unstable imaginary eigenfrequency

FIG. 6. Color online Dependences on C of the real and imaginary parts left and right panels, respectively of eigenfrequencies of small perturbations about vortex solitons with S = 1 in the ordinary nonrotating DNLS lattice top panels , and in the rotating one, with = 0.01 central panels and = 0.04 bottom panels . In the rotating case, the center of the vortex coincides with the rotation pivot of the lattice.

of the S = 2 VS, rendering it real for small C. However, as C is increased the eigenfrequency eventually becomes imaginary again, leading to the instability of the VS.

The stability diagram for the VSs with S = 2 in the C , plane is presented in Fig. 11, indicating the increasing stabilization effect of larger rotation frequencies. As in the case of

S = 1, the instabilities of the VSs with S = 2 transform it into a persistent breather, but without the vortical structure. In fact, a similar qualitative conclusion was made in the ordinary model, with = 0.
C. Quadrupole and octupole solitons

Along with complex solutions for localized vortices with S 2, the ordinary 2D DNLS equation, with = 0, gives rise to real solutions in the form of quadrupoles and octupoles, that resemble higher-order vortices, but carry no topological charge 25,29 . In particular, quadrupoles, which include four lattice sites with alternating phases, have their stability quad in the model with = 0; the same is region 0 C Ccr true for octupoles, which are based on eight sites with alternating phases. In the anticontinuum limit, a case example of the quadrupole and octupole solutions may be seeded, respectively, by the following conﬁgurations: 0,±1 = 1 , ±1,0 = −1, and 0,2 = 1,−1 = 2,1 = −1,0 = 1 , 1,2 = 0,−1 = −1,1 = 2,0 = −1, all other sites having m,n = 0. Note that the latter

FIG. 8. Eigenfrequencies of small perturbations around the vortex with S = 1 are shown for the four different cases: exponential instability for = 0.01 and C = 0.2 top left panel ; linear stability at = 0.01 and C = 0.5 top right panel ; oscillatory instability at = 0.01 and C = 0.8 bottom left panel ; and exponential and oscillatory instability at = 0.05 and C = 0.85 bottom right panel . 046608-9

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1.88 1.86 1.84 1.82 1.8 1.78 1.76 1.74 1.72 0 100 200 300 400 500 600

t

conﬁguration is tantamount to the necklace soliton pattern that was recently observed in a photorefractive crystal with a photoinduced lattice 34 . We have constructed families of quadrupole and octupole soliton solutions in the present model with 0, proceeding from the anticontinuum patterns to ﬁnite C. When doing so, the rotation pivot in Eq. 6 was set at the center of the respective pattern; i.e., we set = = 0 and = = 1 / 2 for the quadrupole and octupole solutions, respectively. It was found quad that the critical values of the coupling constant, Ccr and oct Ccr , which border the stability regions for both these families, very weakly depend on , unlike the situation for the VSs with S = 2, cf. Fig. 11, but quite similar to what was found above for the localized vortices with S = 1 see the right stability border in Fig. 7 .

FIG. 9. Color online The time dependence of the amplitude of a perturbed unstable vortex soliton with S = 1, = 0.03, and C = 0.4. Note that the point corresponding to this soliton is located in the left instability region in Fig. 7, i.e., the vortex is unstable to nonoscillatory perturbations also see the text .

|ψ1,1|

2

V. CONCLUSION

The objective of the present work was to introduce a discrete version of the 2D model combining the self-attractive

FIG. 10. Color online The same as in Fig. 6, but for the vortex solitons with S = 2. Top and bottom panels correspond to = 0 and = 0.05, respectively. Notice the contrast between the absence of instabilities at small C in the bottom right panel and the immediate destabilization in the top right panel. 046608-10

PHYSICAL REVIEW E 76, 046608 2007
S=2 Ccr , as shown in Fig. 11. For the vortices with S = 2, the reverse effect happens in comparison with S = 1, namely, an unstable at = 0 eigenfrequency is tipped by the rotation in the opposite direction; i.e., for S = 1 a real eigenfrequency becomes imaginary in the presence of , while the reverse is true for S = 2. Quadrupole and octupole solitons, with the center coinciding with the pivotal point, were brieﬂy considered too, with a conclusion that their stability regions are almost the same as in the ordinary model with = 0 . An estimate for relevant values of in physical units was given for both physical realizations of the model, i.e., the rotation frequency of the optical lattice stirring the self-attractive BEC, and the pitch of the twisted bundle of optical ﬁbers. The analysis initiated in this work can be developed in several directions. In particular, results obtained in the continuum model considered in Ref. 26 suggest that, at sufﬁciently large , the fully localized FSs, with the center shifted off the axis i.e., continuum counterparts of the discrete FSs considered in the present work , are unstable or do not exist , while stable ring-shaped solitons may appear instead, with the center of the ring coinciding with the pivot. Another interesting direction would be to apply techniques elaborated on in Refs. 37,18 to this considerably more difﬁcult setting. It would be especially relevant to repeat the calculation of the eigenfrequencies for the vortices with S = 1 and S = 2 by means of those methods in the presence of rotation, and quantify the impact of on the eigenvalues, to rigorously investigate some effects which were outlined above in a qualitative form. Another possibility is to consider a discrete limit of the model with the rotating quasi-1D potential, such as the one with the repulsive cubic nonlinearity, which was introduced in Ref. 27 . Studies along these directions are currently underway and will be reported elsewhere.

Ω

C

FIG. 11. Color online The stability diagram, similar to Fig. 7, but for vortex solitons with S = 2.

cubic nonlinearity and rotating square-lattice potential. The discrete model can be implemented in BEC stirred by a rotating strong optical lattice, or, in principle, also in a twisted bundle of nonlinear optical ﬁbers. Localized solutions of two types were considered: off-axis FSs, with the center placed at distance R from the rotation pivot, and on-axis VSs, with vorticity S = 1 and 2. For the FSs, the stability interval was found in the form of 0 C Ccr R , where C is the coupling constant of the discrete lattice, and Ccr R a monotonically decreasing function see Fig. 4 . A qualitative explanation to this result was proposed, based on the analysis of the balance between the lattice-pinning and centrifugal forces. For VSs with S = 1, the dependence of the stability region on the rotation frequency was found, in the form of Eq. 9 and Fig. 7, with the conclusion that the stability is only possible for cr. A key feature, which makes the situation different from earlier stability analyses of such structures in the standard 2D DNLS model, is a higher-order eigenfrequency, shifted toward instability for sufﬁciently weak lattice coupling, in the presence of rotation. On the other hand, VSs with S = 2, which are always unstable in the model with = 0, are stabilized by the rotation in the region 0 C

ACKNOWLEDGMENTS

J.C. acknowledges ﬁnancial support from the MECD Project No. FIS2004-01183. The work of B.A.M. was partially supported by the Israel Science Foundation through Excellence-Center Grant No. 8006/03, and by German-Israel Foundation through Grant No. 149/2006. P.G.K. gratefully acknowledges the support of NSF-DMS-0505663, NSFDMS-0619492, and NSF-CAREER.